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Resolved and subgrid dynamics of Rayleigh–Bénard convection

  • Riccardo Togni (a1), Andrea Cimarelli (a2) and Elisabetta De Angelis (a1) (a2)


In this work we present and demonstrate the reliability of a theoretical framework for the study of thermally driven turbulence. It consists of scale-by-scale budget equations for the second-order velocity and temperature structure functions and their limiting cases, represented by the turbulent kinetic energy and temperature variance budgets. This framework represents an extension of the classical Kolmogorov and Yaglom equations to inhomogeneous and anisotropic flows, and allows for a novel assessment of the turbulent processes occurring at different scales and locations in the fluid domain. Two relevant characteristic scales, $\ell _{c}^{u}$ for the velocity field and $\ell _{c}^{\unicode[STIX]{x1D703}}$ for the temperature field, are identified. These variables separate the space of scales into a quasi-homogeneous range, characterized by turbulent kinetic energy and temperature variance cascades towards dissipation, and an inhomogeneity-dominated range, where the production and the transport in physical space are important. This theoretical framework is then extended to the context of large-eddy simulation to quantify the effect of a low-pass filtering operation on both resolved and subgrid dynamics of turbulent Rayleigh–Bénard convection. It consists of single-point and scale-by-scale budget equations for the filtered velocity and temperature fields. To evaluate the effect of the filter length $\ell _{F}$ on the resolved and subgrid dynamics, the velocity and temperature fields obtained from a direct numerical simulation are split into filtered and residual components using a spectral cutoff filter. It is found that when $\ell _{F}$ is smaller than the minimum values of the cross-over scales given by $\ell _{c,min}^{\unicode[STIX]{x1D703}\ast }=\ell _{c,min}^{\unicode[STIX]{x1D703}}Nu/H=0.8$ , the resolved processes correspond to the exact ones, except for a depletion of viscous and thermal dissipations, and the only role of the subgrid scales is to drain turbulent kinetic energy and temperature variance to dissipate them. On the other hand, the resolved dynamics is much poorer in the near-wall region and the effects of the subgrid scales are more complex for filter lengths of the order of $\ell _{F}\approx 3\ell _{c,min}^{\unicode[STIX]{x1D703}}$ or larger. This study suggests that classic eddy-viscosity/diffusivity models employed in large-eddy simulation may suffer from some limitations for large filter lengths, and that alternative closures should be considered to account for the inhomogeneous processes at subgrid level. Moreover, the theoretical framework based on the filtered Kolmogorov and Yaglom equations may represent a valuable tool for future assessments of the subgrid-scale models.


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Ahlers, G., Bodenschatz, E., Funfschilling, D., Grossmann, S., He, X., Lohse, D., Stevens, R. J. A. M. & Verzicco, R. 2012 Logarithmic temperature profiles in turbulent Rayleigh–Bénard convection. Phys. Rev. Lett. 109 (11), 114501.10.1103/PhysRevLett.109.114501
Antonia, R. A. & Orlandi, P. 2003 Effect of Schmidt number on small-scale passive scalar turbulence. Appl. Mech. Rev. 56 (6), 615632.10.1115/1.1581885
Bryan, G. H., Wyngaard, J. C. & Fritsch, J. M. 2003 Resolution requirements for the simulation of deep moist convection. Mon. Weath. Rev. 131 (10), 23942416.10.1175/1520-0493(2003)131<2394:RRFTSO>2.0.CO;2
Burattini, P., Antonia, R. A. & Danaila, L. 2005 Scale-by-scale energy budget on the axis of a turbulent round jet. J. Turbul. (6), N19.10.1080/14685240500213744
Cabot, W. H. 1993 Large eddy simulations of time-dependent and buoyancy-driven channel flows. Annual Research Briefs 1992. pp. 4560. CTR Annual Research Briefs (Center for Turbulence Research, Stanford University/NASA Ames).
Chillà, F. & Schumacher, J. 2012 New perspectives in turbulent Rayleigh–Bénard convection. Eur. Phys. J. E 35 (7), 125.
Cimarelli, A. & De Angelis, E. 2011 Analysis of the Kolmogorov equation for filtered wall-turbulent flows. J. Fluid Mech. 676, 376395.10.1017/S0022112011000565
Cimarelli, A. & De Angelis, E. 2012 Anisotropic dynamics and sub-grid energy transfer in wall-turbulence. Phys. Fluids 24 (1), 015102.10.1063/1.3675626
Cimarelli, A. & De Angelis, E. 2014 The physics of energy transfer toward improved subgrid-scale models. Phys. Fluids 26 (5), 055103.10.1063/1.4871902
Cimarelli, A., De Angelis, E. & Casciola, C. M. 2013 Paths of energy in turbulent channel flows. J. Fluid Mech. 715, 436451.10.1017/jfm.2012.528
Cimarelli, A., De Angelis, E., Jiménez, J. & Casciola, C. M. 2016 Cascades and wall-normal fluxes in turbulent channel flows. J. Fluid Mech. 796, 417436.10.1017/jfm.2016.275
Corrsin, S. 1951 On the spectrum of isotropic temperature fluctuations in an isotropic turbulence. J. Appl. Phys. 22 (4), 469473.10.1063/1.1699986
Dabbagh, F., Trias, F. X., Gorobets, A. & Oliva, A. 2016 New subgrid-scale models for large-eddy simulation of Rayleigh–Bénard convection. J. Phys. Conf. Ser. 745 (3), 032041.10.1088/1742-6596/745/3/032041
Dabbagh, F., Trias, F. X., Gorobets, A. & Oliva, A. 2017 A priori study of subgrid-scale features in turbulent Rayleigh–Bénard convection. Phys. Fluids 29 (10), 105103.10.1063/1.5005842
Danaila, L., Anselmet, F., Zhou, T. & Antonia, R. A. 2001 Turbulent energy scale budget equations in a fully developed channel flow. J. Fluid Mech. 430, 87109.10.1017/S0022112000002767
Danaila, L., Krawczynski, J. F., Thiesset, F. & Renou, B. 2012 Yaglom-like equation in axisymmetric anisotropic turbulence. Physica D 241 (3), 216223.10.1016/j.physd.2011.08.011
Davidson, P. A., Pearson, B. R. & Staplehurst, P. 2004 How to describe turbulent energy distributions without the Fourier transform. In Proceedings of the AFMC 15, Sydney, Australia. University of Sydney.
Deardorff, J. W. 1974 Three-dimensional numerical study of the height and mean structure of a heated planetary boundary layer. Boundary-Layer Meteorol. 7 (1), 81106.10.1007/BF00224974
Domaradzki, J. A., Liu, W., Härtel, C. & Kleiser, L. 1994 Energy transfer in numerically simulated wall-bounded turbulent flows. Phys. Fluids 6, 15831599.10.1063/1.868272
Dupuy, D., Toutant, A. & Bataille, F. 2018 Turbulence kinetic energy exchanges in flows with highly variable fluid properties. J. Fluid Mech. 834, 554.10.1017/jfm.2017.729
Gauding, M., Wick, A., Pitsch, H. & Peters, N. 2014 Generalised scale-by-scale energy-budget equations and large-eddy simulations of anisotropic scalar turbulence at various Schmidt numbers. J. Turbul. 15 (12), 857882.10.1080/14685248.2014.935385
Gayen, B., Hughes, G. O. & Griffiths, R. W. 2013 Completing the mechanical energy pathways in turbulent Rayleigh–Benard convection. Phys. Rev. Lett. 111 (12), 124301.10.1103/PhysRevLett.111.124301
Grossmann, S. & Lohse, D. 2000 Scaling in thermal convection: a unifying theory. J. Fluid Mech. 407, 2756.10.1017/S0022112099007545
Härtel, C., Kleiser, L., Unger, F. & Friedrich, R. 1994 Subgrid-scale energy transfer in the near-wall region of turbulent flows. Phys. Fluids 6 (9), 31303143.10.1063/1.868137
Hill, R. J. 2002 Exact second-order structure–function relationships. J. Fluid Mech. 468, 317326.10.1017/S0022112002001696
Kimmel, S. J. & Domaradzki, J. A. 2000 Large eddy simulations of Rayleigh–Bénard convection using subgrid scale estimation model. Phys. Fluids 12 (1), 169184.10.1063/1.870292
Kolmogorov, A. N. 1941a Dissipation of energy in locally isotropic turbulence. Dokl. Akad. Nauk SSSR 32, 1618.
Kolmogorov, A. N. 1941b The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers. In Dokl. Akad. Nauk SSSR, vol. 30, pp. 301305. JSTOR.
Lilly, D. K. 1962 On the numerical simulation of buoyant convection. Tellus 14 (2), 148172.10.3402/tellusa.v14i2.9537
Lilly, D. K. 1967 The representation of small scale turbulence in numerical simulation experiments. In Proceedings of the IBM Scientific Computing Symposium on Environmental Science, pp. 195210. IBM.
Liu, S., Meneveau, C. & Katz, J. 1994 On the properties of similarity subgrid-scale models as deduced from measurements in a turbulent jet. J. Fluid Mech. 275, 83119.10.1017/S0022112094002296
Lohse, D. & Xia, K.-Q. 2010 Small-scale properties of turbulent Rayleigh–Bénard convection. Annu. Rev. Fluid Mech. 42, 335364.10.1146/annurev.fluid.010908.165152
Marati, N., Casciola, C. M. & Piva, R. 2004 Energy cascade and spatial fluxes in wall turbulence. J. Fluid Mech. 521, 191215.10.1017/S0022112004001818
Mason, P. J. 1989 Large-eddy simulation of the convective atmospheric boundary layer. J. Atmos. Sci. 46 (11), 14921516.10.1175/1520-0469(1989)046<1492:LESOTC>2.0.CO;2
Obukhov, A. M.1968 Structure of the temperature field in turbulent flow. Tech. Rep. DTIC Document.
Piomelli, U. 1999 Large-eddy simulation: achievements and challenges. Prog. Aerosp. Sci. 35 (4), 335362.10.1016/S0376-0421(98)00014-1
Piomelli, U. & Balaras, E. 2002 Wall-layer models for large-eddy simulations. Annu. Rev. Fluid Mech. 34 (1), 349374.10.1146/annurev.fluid.34.082901.144919
Piomelli, U., Cabot, W. H., Moin, P. & Lee, S. 1991 Subgrid-scale backscatter in turbulent and transitional flows. Phys. Fluids A 3 (7), 17661771.10.1063/1.857956
Pope, S. B.2001 Turbulent Flows. Cambridge University Press.
Porté-Agel, F., Parlange, M. B., Meneveau, C. & Eichinger, W. E. 2001 A priori field study of the subgrid-scale heat fluxes and dissipation in the atmospheric surface layer. J. Atmos. Sci. 58 (18), 26732698.10.1175/1520-0469(2001)058<2673:APFSOT>2.0.CO;2
Sergent, A., Joubert, P. & Le Quéré, P. 2006 Large-eddy simulation of turbulent thermal convection using a mixed scale diffusivity model. Prog. Comput. Fluid Dyn. 6 (1-3), 4049.10.1504/PCFD.2006.009481
Siggia, E. D. 1994 High Rayleigh number convection. Annu. Rev. Fluid Mech. 26 (1), 137168.10.1146/annurev.fl.26.010194.001033
Smagorinsky, J. 1963 General circulation experiments with the primitive equations. 1. The basic experiment. Mon. Weath. Rev. 91 (3), 99164.10.1175/1520-0493(1963)091<0099:GCEWTP>2.3.CO;2
Togni, R., Cimarelli, A. & De Angelis, E. 2015 Physical and scale-by-scale analysis of Rayleigh–Bénard convection. J. Fluid Mech. 782, 380404.10.1017/jfm.2015.547
Valente, P. C. & Vassilicos, J. C. 2015 The energy cascade in grid-generated non-equilibrium decaying turbulence. Phys. Fluids 27 (4), 045103.10.1063/1.4916628
Van Reeuwijk, M., Jonker, H. J. J. & Hanjalić, K. 2005 Identification of the wind in Rayleigh–Bénard convection. Phys. Fluids 17 (5), 051704.10.1063/1.1920350
Yaglom, A. M. 1949 On the local structure of a temperature field in a turbulent flow. Dokl. Akad. Nauk SSSR 69 (6), 743.
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